A sprinter can accelerate with constant acceleration for 2.10s before reaching top speed. He can run the 100-meter dash in 10 s.
What is his speed as he crosses the finish line?
To find the sprinter's speed as he crosses the finish line, we need to use the equation of motion:
\(v = u + at\)
Where:
v = final velocity (speed)
u = initial velocity (speed)
a = acceleration
t = time
Since the sprinter accelerates for 2.10 seconds, the initial velocity will be zero (as he starts from rest at the beginning of the race).
Let's assume that the sprinter accelerates uniformly.
First, let's find the acceleration (a).
Since we know the time it takes for the sprinter to accelerate, we can use the equation:
\(a = \frac{{v - u}}{{t}}\)
Where:
a = acceleration
v = final velocity (which we need to find)
u = initial velocity (0 m/s)
t = time (2.10 s)
Substituting the given values into the equation:
\(a = \frac{{v - 0}}{{2.10}}\)
\(a = \frac{{v}}{{2.10}}\)
Simplifying the equation, we have:
\(2.10a = v\)
Now, we need to find the final velocity (v) when the sprinter crosses the finish line (after 10 seconds).
Using the equation of motion:
\(v = u + at\)
Where:
u = initial velocity (0 m/s)
a = acceleration (which we found to be 2.10a)
t = time (10 s)
Substituting the given values into the equation:
\(v = 0 + (2.10a)(10)\)
\(v = 21a\)
Since we already found that \(2.10a = v\), we can substitute this value into the equation:
\(v = 21(2.10a)\)
\(v = 44.1a\)
Therefore, the sprinter's speed as he crosses the finish line is 44.1a m/s.
To find the speed of the sprinter as he crosses the finish line, we can use the concept of constant acceleration.
First, let's determine the acceleration of the sprinter during the first 2.10 seconds. We're given that the acceleration is constant, so we can use the formula:
acceleration = change in velocity / time
We assume the sprinter starts from rest (zero velocity). Let's say the sprinter reaches his top speed (v) after accelerating for 2.10 seconds. So, the change in velocity is simply the top speed:
change in velocity = top speed - initial velocity = v - 0 = v
Substituting the values into the formula, we have:
acceleration = v / 2.10
Now, we need to find what this acceleration is. Since the distance covered during the acceleration phase is not mentioned, we cannot directly calculate the acceleration. However, we know that after the 2.10 seconds of acceleration, the sprinter reaches his top speed and completes the 100-meter dash.
Given the sprinter completes the 100-meter dash in 10 seconds, we can use this information to calculate the time they spend reaching top speed:
time = total time - time spent on acceleration = 10 s - 2.10 s = 7.90 s
Now, we can use the formula for constant velocity to calculate the top speed:
velocity = acceleration * time
Substituting the values, we have:
v = acceleration * 7.90
Substituting the expression for acceleration from earlier, we get:
v = (v / 2.10) * 7.90
Now, let's solve this equation for v:
v = (v * 7.90) / 2.10
Multiplying both sides by 2.10 to get rid of the denominator:
2.10v = v * 7.90
Dividing both sides by v to isolate v:
2.10 = 7.90
Since the equation simplifies to 2.10 = 7.90, the solution for v is 0. This means there is an error in the problem statement or calculation.
Therefore, using the given information, we cannot determine the speed of the sprinter as he crosses the finish line.
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the runner runs at top speed for 7.9 seconds, so
1/2 a*2.1^2 + 2.1a*7.9 = 100
a=5.32
v = at = 5.32*2.1 = 11.172 m/s
check:
accelerating at 5.32m/s^2 for 2.1 s gives
s = 1/2 * 5.32 * 2.1^2 = 11.73
From then on with speed = 5.32*2.1 = 11.172, the distance covered is
11.172*7.9 = 88.27
total distance: 100.0 m